Question: Multiply the following complex numbers, marked as blue dots on the graph: $[4(\cos(\frac{7}{12}\pi) + i \sin(\frac{7}{12}\pi))] \cdot [\cos(\frac{7}{4}\pi) + i \sin(\frac{7}{4}\pi)]$ (Your current answer will be plotted in orange.)
Answer: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $4(\cos(\frac{7}{12}\pi) + i \sin(\frac{7}{12}\pi))$ ) has angle $\frac{7}{12}\pi$ and radius $4$ The second number ( $\cos(\frac{7}{4}\pi) + i \sin(\frac{7}{4}\pi)$ ) has angle $\frac{7}{4}\pi$ and radius $1$ The radius of the result will be $4 \cdot 1$ , which is $4$ The sum of the angles is $\frac{7}{12}\pi + \frac{7}{4}\pi = \frac{7}{3}\pi$ The angle $\frac{7}{3}\pi$ is more than $2 \pi$ . A complex number goes a full circle if its angle is increased by $2 \pi$ , so it goes back to itself. Because of that, angles of complex numbers are convenient to keep between $0$ and $2 \pi$ $\frac{7}{3}\pi - 2 \pi = \frac{1}{3}\pi$ The radius of the result is $4$ and the angle of the result is $\frac{1}{3}\pi$.